Category of modules information

ring R, the category of left modules over R is the category whose objects are all left modules over R and whose morphisms are all module homomorphisms...

entire category of modules. Injective resolutions measure how far from injective a module is in terms of the injective dimension and represent modules in...

ring, then the category of finitely generated left modules over R is abelian. In particular, the category of finitely generated modules over a noetherian...

class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, keeping some of the main properties of free...

sheaf of O-modules or simply an O-module over a ringed space (X, O) is a sheaf F such that, for any open subset U of X, F(U) is an O(U)-module and the...

one can consider a category C enriched over the monoidal category of modules over a commutative ring R, called an R-linear category. In other words, each...

{\mathcal {A}}} be an abelian category. (Examples include the category of modules over a ring and the category of sheaves of abelian groups on a topological...

combines several modules into a new module. The most familiar examples of this construction occur when considering vector spaces, which are modules over a field...

(the monoidal category of R-modules). By definition, a ring is a monoid object in the category of abelian groups; thus, the notion of an associative...

spaces, abelian groups, R-modules, or R-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples...

particular, any non-trivial category with a zero object, such as an abelian category, is not Cartesian closed. So the category of modules over a ring is not Cartesian...

let R-Mod be the category of modules over R. (One can take this to mean either left R-modules or right R-modules.) For a fixed R-module A, let T(B) = HomR(A...

algebra, a module homomorphism is a function between modules that preserves the module structures. Explicitly, if M and N are left modules over a ring...

representation theory, the stable module category is a category in which projectives are "factored out." Let R be a ring. For two modules M and N over R, define...

compact modules, also called pure-injective modules, are modules that have a certain "nice" property which allows the solution of infinite systems of equations...

A} , there is an equivalence of categories from A {\displaystyle A} -modules to quasi-coherent sheaves, taking a module M {\displaystyle M} to the associated...

category C {\displaystyle {\mathcal {C}}} , and the category of modules over the ring is a category of functors defined on C {\displaystyle {\mathcal {C}}}...

over R, or a module of finite type. Related concepts include finitely cogenerated modules, finitely presented modules, finitely related modules and coherent...

regarded as Z-modules, so the category of abelian groups is also a symmetric, closed monoidal category. A symmetric compact closed category is a symmetric...

related to R-modules, but where this map is an isomorphism turns out to be the category of R [ S − 1 ] {\displaystyle R[S^{-1}]} -modules. Here R [ S −...

of the ∞-category of perfect modules over R (a perfect module being defined as a compact object in the ∞-category of module spectra.) G-spectrum J. Lurie...

combines several modules into a new, larger module. The direct sum of modules is the smallest module which contains the given modules as submodules with...

similar notion of duality in 1958. Rings are commonly studied in terms of their modules, as modules can be viewed as representations of rings. Every ring...

R-Mod the category of left R-modules and by Mod-R the category of right R-modules (if R is commutative, the two categories coincide). Fix a module B in R-Mod...

states that any small abelian category can be represented as a category of modules over some ring. For the category of groups, just turn all additive...